The downside of this approach is that it suffers from “double asymptotics". We run one MCMC chain generating samples from \(\pi ^{(k)}_{\delta }(\varphi |Y,Z)\) . For each sample \(\varphi \)  output in this run we simulate a chain targeting \(\pi (\theta |Y,\varphi )\)  and take the last sampled \(\theta \) -value. This second chain must run to convergence. Whilst in our experience very high accuracy can be achieved in a modest runtime, of the order of ten times the runtime of the chain targeting the Bayes posterior \(\pi (\varphi ,\theta |Y,Z)\)  for the same ESS [1]}, this is clearly a weakness of this scheme. It may be preferable to analyse the \(\delta \) -SMI posterior using the variational framework of [2]} and [3]}.
